Large Banking Systems with Default and Recovery: A Mean Field Game Model
ROMUALD ELIE ∗ TOMOYUKI ICHIBA † MATHIEU LAURIERE ‡
Abstract We consider a mean-field model for large banking systems, which takes into account default and recovery of the institutions. Building on models used for groups of interacting neurons, we first study a McKean-Vlasov dynamics and its evolutionary Fokker-Planck equation in which the mean-field interactions occur through a mean-reverting term and through a hitting time corresponding to a default level. The latter feature reflects the impact of a financial institution’s default on the global distribution of reserves in the banking system. The systemic risk problem of financial institutions is understood as a blow-up phenomenon of the Fokker-Planck equation. Then, we incorporate in the model an optimization component by letting the institutions control part of their dynamics in order to minimize their expected risk. Phrasing this optimization problem as a mean-field game, we provide an explicit solution in a special case and, in the general case, we report numerical experiments based on a finite difference scheme.
1 Introduction
Financial institutions form a highly connected network through monetary flow and complex de- pendencies. Each institution is trying to maximize its expected return objective over time, while the aggregation of all investment strategies generates feedback loops and results in some overall patterns of the financial market. The institutions are all competing against each other as players in a financial interacting game. When the number of players becomes large as we can observe in the current fully connected worldwide banking system, individual interactions become intractable while the global patterns become more apparent. Our goal in this paper is to capture the origins of these patterns in a simple mathematical and numerical setup, described by a mean-field game banking system, taking into account births and defaults of financial institutions. The set up of the birth and default dynamics considered here is inspired by the neuron firing arXiv:2001.10206v1 [math.OC] 28 Jan 2020 models [3, 9, 8, 17] as well as mathematical physics models [25, 26]; see also [12, 13, 19, 23]. These models aim at describing how the electric potential of a network of neurons evolves over time. A salient feature occurs whenever the potential of one single neuron reaches a given threshold level, leading to a so-called firing event where all the energy accumulated is transmitted to neighboring neurons in the network. In these models, it has been observed that a few parameters characterize some sort of phase transition period, according to which steady states or blow-up phenomena may occur or not. A blow-up typically corresponds to the situation in which the firing of one neuron
∗ Université Paris-Est Marne-la-Vallée, France (E-mail: [email protected]). † Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA 93106, USA (E-mail: [email protected]). Research supported in part by the National Science Foundation under grants NSF-DMS-13-13373 and DMS-1615229 ‡ Department of Operations Research and Financial Engineering, Sherrerd Hall, Princeton University, Princeton, NJ 08540, USA (E-mail: [email protected]).
1 immediately triggers the firing of some other neurons and so on, resulting in a cascade of firings. In the macroscopic limit, this translates into a jump in the instantaneous rate of firings. Handling in a mathematically rigorous way such phenomenon is an active and very challenging research area, in which recent progress has been made for instance in [10] in connection with the aforementioned line of work. Drawing an analogy between neurons firing and banks defaulting, the blow-up phenomenon appears as a natural tool for describing systemic patterns of a financial crisis: complicated interac- tions between poorly regulated banks driven by selfish objectives can sometimes lead to cascade of defaults. This line of modeling has already been pointed out e.g. in [7, 26] to describe the evolution of a financial system. Building on this type of model, our main contribution in this paper is to incorporate a proper optimization component for each bank, by letting each bank influence the dynamics of its state so as to minimize a chosen idiosyncratic risk criterion, in the spirit of [6].
More specifically, we focus here on the mean-field limit of the following toy model: The banking system consists of N banks, where each bank i owns liquid assets as monetary reserve evolving i continuously in time. Let denote by (Xt )t≥0 the non-negative level of cash reserve (in liquid assets) of bank i , for i = 1,...,N and introduce (Xt)t≥0 the average level of cash reserves over the full 1 N banking system, i.e., Xt := (Xt + ··· + Xt ) /N at time t ≥ 0 . We assume for simplicity that the diffusion dynamics of each cash reserve Xi is of mean field type, i.e. it depends on the global banking system through the empirical distribution of the cash reserve levels, and more specifically through their empirical average X . More precisely, the diffusive i 2 i behavior of X at time t is locally specified by a drift function b : R+ → R of its value Xt together i with the running average Xt of the system. The sample path t → Xt is assumed to be right continuous with left limits. i0 The bank defaults are modeled in the following way: if the amount Xt of liquid assets of bank i0 reaches the threshold level 0 at time t0 , this bank i0 is defaulted from the system. A systemic effect induces a financial shock to each institution of the system, so that the cash reserve Xj of every bank j(=6 i0) suffers at time t0− a downward jump of size Xt0−/N . Instantaneously and for ease of modeling, a new institution is also created in the market with cash reserve at the average level Xt0−, so that the number of banks in the financial system remains constant. For notational simplicity, this new bank keeps the same number , i.e., i0 i0 . With initial i0 Xt0− = 0 ,Xt0+ := Xt0 1 N N configuration x0 := (X0 ,...,X0 ) ∈ (0, ∞) , the resulting dynamics may be depicted by the following system of stochastic differential equations
Z t Z t i i i i i 1 X j X = X + b(X , Xs)ds + σW + Xs− dM − dM ; t ≥ 0 , t 0 s t s N s 0 0 j6=i ∞ (1) i X i n i i Xs− X j j o Mt := 1{τ i ≤t} , τk := inf s > τk−1 : Xs− − (Ms − Ms−) ≤ 0 ; k ∈ N , k N k=1 j6=i for i = 1,...,N , where W := (W 1,...,W N ) , t ≥ 0 is a standard N-dimensional Brownian mo- 1 N i tion, X is the average of X := (X ,...,X ) , Mt is the cumulative number of defaults of bank i i i by time t ≥ 0 , and τk is its k -th default time with τ0 = 0 . Although, in our toy model, the drift coefficient b depends on both the state X of a bank together with the average X whereas the dif- fusion coefficient is a positive constant σ, one can in general consider more complicated dependence on the coefficients, such as time-dependent drift and diffusion coefficients. The system (1) of N banks includes mean-field interactions through the drift function b(·) as well as via the cumulative number of defaults M. The mathematical analysis of such system raises some delicate and technical
2 issues, due to mean-field interactions together with multiple defaults.
We then turn our attention to the more realistic situation in which banks are not passively following the dynamics (1) but are able to partly control their drifts and seek to minimize the sum of expected running costs (possibly together with a terminal default cost), occurring at default time. Compared with the model (1), the drift of bank i incorporates a linear dependence on the control i ξt used by this bank. The running cost rate f at time t ≥ 0 of bank i depends on the control i i PN ξ , the monetary reserve X as well as the empirical distribution mN,t(·) := δ i (·) /N of t t i=1 Xt the system. This allows in particular to penalize strong deviations from the average of the banking system. Here, we borrow the running cost functional from a model introduced in [6] for systemic risk: The running cost functional takes the form of a quadratic function. Moreover, the running cost is discounted at rate r . Thus, over the time interval [0,T ] , each bank i seeks to minimize
"Z τ i∧T # −rs i i E e f(Xs, mN,s, ξs)ds , 0 where X is controlled by ξ via its drift. In this context, we look for a Nash equilibrium, in the sense that when bank i performs the optimization, the controls used by the other banks j =6 i are fixed. This modeling of systemic risk hence identifies to an N -player stochastic game, as the other players policy and default rate modify the banking system dynamics, under which each bank optimizes its policy and resulting monetary reserve. As the size of the system becomes large, i.e., N → ∞ , the difficulty and complexity of the mathematical analysis increase rapidly. In order to obtain a tractable approximation of a large banking system, we rely on the mean field game (MFG) paradigm. Mean field games were introduced by Lasry and Lions [24, 22, 20, 21] and Caines, Huang and Malhamé [14, 15, 16]. The interested reader is referred to the recently published books [2, 4, 5] and the references therein.
The rest of the paper is organized as follows. In section 2, we first show that the N-agent system (1) is well defined using a notion of physical solution borrowed from [9]. We then informally derive the limiting mean-field system as N → ∞ , as well as the corresponding nonlinear Fokker- Plank equation. In the limiting system, we show in Theorem 4 that blow-up may occur when the initial distribution is concentrated near the origin. This result shades in particular a new light on the blow-up phenomenon described in [3]. Importantly, we observe that the Fokker-Plank system has an explicit stationary solution, as derived in Theorem 5. We conclude section 2 with numerical results for the dynamics of the system. In section 3, we incorporate the optimization component. We formulate the mean-field game in section 3.1. Its solution is characterized by a Hamilton-Jacobi- Bellman (HJB) equation coupled with a Fokker Plank (FP) equation with well suited boundary conditions at the boundary of the domain. In section 3.3, we derive an explicit solution for a stationary mean field game from the PDE system, where Theorem 5 is used to determine the probability distribution. This stationary solution serves as a baseline for our numerical study. In section 3.4, we provide a discrete numerical scheme for the HJB-FP system, building on [1]. Finally, numerical results in a non-stationary regime are presented in section 3.5.
3 2 Fokker Planck equation for the particle system
2.1 Construction of Physical Solution 2.1.1 Modeling bank interactions and defaults Throughout this section, we assume for simplicity that σ = 1, and in this subsection, let us assume 2 that b(·, ·) in (1) is (globally) Lipschitz continuous on R+ , i.e., there exists a constant κ > 0 such that |b(x1, m1) − b(x2, m2)| ≤ κ |x1 − x2| + |m1 − m2| (2) for all x1, x2, m1, m2 ∈ R+ , and impose the following condition on the drift function b(·, ·) :
N X b(xi, x) ≡ 0 (3) i=1
1 N N 1 N for every x := (x , . . . , x ) ∈ R+ and x := (x + ··· + x ) /N . This condition holds for instance if b(xi, x) = xi − x is a linear mean-reverting drift, and naturally translates a global stability of the monetary level for the whole financial system.
1 N Given a standard Brownian motion W· , we shall consider a system (X· := (X· ,...,X· ),M· := 1 N (M· ,...,M· )) described by (1) together with conditions (2)-(3) on a filtered probability space (Ω, F, F, P) , with filtration F := (Ft, t ≥ 0) . In particular, we are concerned with identifying cases where dynamics (1) might induce multiple simultaneous defaults with positive probability, i.e.,
i j P ∃(i, j), ∃t ∈ [0, ∞) such that Xt = Xt = 0 > 0 .
In the dynamics (1) of X· the last term containing the counting process M· describes how those banks behave at the default event times. In principle, there are other specifications of their behaviors. For example, replacing 1/N by 1/(N − 1) in (1), we observe the dynamics of X· = 1 N (X· ,...,X· ) becomes
i i i i 1 X j dX = b(X , Xt)dt + dW + Xt− dM − dM (4) t t t t N − 1 t j6=i for i = 1,...,N , t ≥ 0 , and by direct calculations we may verify that under (3) the average PN i process X· of X· in (4) moves as a Brownian motion X· = X0 + (1 /N) i=1 W· . If we consider a (weak) solution on a probability space (Ω, F, F, P) of the system X· with dynamics (4) which N takes values in [0, ∞) only, then the first passage time inf{t ≥ 0 : X(t) = 0} of 0 for X· is finite almost surely and hence, the N -dimensional process X· in (4) hits the origin almost surely under (3), i.e., 1 N P ∃t ∈ [0, ∞) such that Xt = ··· = Xt = 0 = 1 . (5) Another example with such property occurs with the following dynamics for i = 1,...,N
N i i i i 1 X j dX = b(X , Xt)dt + dW + Xt− dM − dM (6) t t t t N t j=1 with (3). Again, this dynamics implies that the average process X· is a Brownian motion, and hence the N -dimensional process X· in (6) entails the property (5) with the above reasoning.
4 On the other hand, in the dynamics (1) the average process X· jumps up at the default event times as we observe
N N Z · Z · 1 X i 1 X i 1 X· = X0 + W + Xs−dM = X0 + W · + Xs−dM s, (7) N · N 2 s N i=1 i=1 0 0 where M · and W · are the sample averages of M· and W· , respectively. In between default times, X· behaves as the N -dimensional diffusion with drift b(·, ·) and unit diffusion coefficients. When i there is a jump (default) in the process (X·,M·) at time t , the jump size Xt of defaulted bank i is given by
i i i i 1 X j j X − X = Xt− · M − M − M − M ; i = 1,...,N. (8) t t− t t− N t t− j6=i Economically speaking, in case of defaults, the creation of a new financial institution requires additional funding from another global financial agency (e.g., government) outside the system.
2.1.2 Connections with the Neuron firing model of Delarue et al. [8, 9] Up to a well chosen transformation, we now observe that the dynamics of the financial banking system considered here, shares some similarity with the dynamics derived in the firing Neuronal systems of Delarue et al. [8, 9]. N Note that the value of Xt lies in the state space [0, ∞) for each t ≥ 0 . Thus if we change N N 1 N the state space from [0, ∞) to (−∞, 1] by transforming Xt to Xbt := (Xbt ,..., Xbt ) with i i PN i Xb· := (X· − X· ) / X· , i = 1,...,N , then we see i=1 Xbt ≡ 0 and by Itô’s rule the dynamics of Xb· is given by
i i i N b(Xt(1 − Xbt ), Xt) Xbt 1 1 − Xbt− 1 X j dXbi = − + dt+dWci− 1+ dMci+ 1+ dMc , (9) t 2 t N t N N t Xt NXt j=1
i N ∞ i 1 i 1 − Xbt 1 X j i X i dWct := − dWt + dWt , Mct := 1{τ i ≤t} ≡ Mt , N bk Xt Xt j=1 k=1
i n i i 1 X j j o i i τk := inf t > τk−1 : Xbt + Mcs − Mcs− ≥ 1 = τk , τ0 ≡ 0 ; i = 1, . . . , N , k ∈ N b b N b j6=i until the time inf{t : Xt = 0} . 1 N This system (Xb· ,..., Xb· , X·) defined by (7) and (9) resembles with the particle system Xet := 1 N (Xet ,..., Xet ) for neurons studied in [8, 9], namely,
N α X j dXei = b(Xei)dt + dWfi − dMfi + dMf (10) t t t t N t j=1 for i = 1, . . . , N, t ≥ 0 in modeling a very large network of interacting spiking neurons. Here 1 N b : (−∞, 1] → R is a Lipschitz continuous function, Wft := (Wft ,..., Wft ) , t ≥ 0 is the standard Brownian motion and
∞ N i X i n i α X j j o Mft := 1{τ i ≤t} , τk := inf s > τk−1 : Xes− + (Mfs − Mfs−) ≥ 1 ; k ∈ N ek e e N k=1 j=1
5 i with τe0 = 0 , i = 1,...,N for t ≥ 0 . It is known that if the parameter α lies in (0, 1) , there exists a unique solution (called “physical solution”) to (10), such that
i i α |Γet| i i α |Γet| Xe = X + if i 6∈ Γet , Xe = X + − 1 if i ∈ Γet , t t− N t t− N where |Γet| is the cardinality of Γet , a random subset of indexes {1,...,N} defined by the union [ Γet := Γet,k , 0≤k≤N−1
i of recursively defined sets Γet,0 := {i ∈ {1,...,N} : Xt− = 1} ,
k k n [ i α [ o Γet,k+1 := i ∈ {1,...,N}\ Γet,` : X + Γet,k ≥ 1 , t− N `=0 `=0 for k = 0, 1,...,N − 2 , t ≥ 0 . Again here | · | represents the cardinality of set.
2.1.3 Physical solutions In a similar spirit, we shall construct a solution to (1) with a specific boundary behavior at default times. Let us define the following map Φ(x) := (Φ1(x),..., ΦN (x)) : [0, ∞)N 7→ [0, ∞)N and N i set-valued function Γ: R+ → {1,...,N} defined by Γ0(x) := {i ∈ {1,...,N} : x = 0} , k k n [ x [ o Γ (x) := i ∈ {1,...,N}\ Γ (x): xi − · Γ (x) ≤ 0 ; k = 0, 1, 2,...,N − 3 k+1 ` N ` `=1 `=1
N−2 [ 1 1 Γ(x) := Γ (x) , Φi(x) := xi + x 1 + · 1 − · |Γ(x)| (11) k N {i∈Γ(x)} N k=0 1 N N 1 for x = (x , . . . , x ) ∈ R+ , i = 1,...,N with x := (x + ··· + xN ) /N ≥ 0 . Note that Φ([0, ∞)N \{0}) ⊆ [0, ∞)N \{0} and Φ(0) = 0 = (0,..., 0) .
1 N N Given the initial configuration X0 := (X0 ,...,X0 ) ∈ (0, ∞) and a standard N-dimensional 1 1,1 1,N Brownian motion W , we take the unique strong solution Y· := (Y· ,...,Y· ) to Z t 1,i i 1,i 1 i Yt = X0 + b(Ys , Y s) ds + Wt ; i = 1, . . . , N , t ≥ 0 , (12) 0
1 1 N thanks to the Lipschitz continuity of b(·, ·) as in (2). Here Y · := (Y· + ··· + Y· ) /N . Let us define for 0 ≤ t < τ 1 i 1,i i Xt := Yt ,Mt := 0 , (13) 1 1,i 1 where τ := min1≤i≤N inf{s ≥ 0 : Ys = 0} , and at τ let us define i i 1 i Xτ 1 := Φ (Yτ 1 ) ,Mτ 1 := 1{ i∈Γ(Y 1 ) } , τ1 N where the map Φ(·) and Γ(·) are defined in (11). Then, whenever Xτ k ∈ [0, ∞) \{0} for k+1 k+1,1 k+1,N k = 1, 2,... , we construct recursively the unique strong solution Y· := (Y· ,...,Y· ) to the system of stochastic differential equations Z t k+1,i i k+1,i k+1 i Yt = Xτ k + b(Ys , Y s ) ds + Wt ; i = 1, . . . , N , t ≥ 0 , (14) 0
6 k+1 k+1 where Y · is the average of elements of Y· , and define k+1 k k+1,i i k+1,i i i k k+1 τ := min inf{s ≥ τ : Ys = 0} ,Xt := Yt ,Mt := M k for τ ≤ t < τ , 1≤i≤N τ
i i k+1,i i i Xτ k+1 := Φ (Yτ k+1 ) ,Mτ k+1 := Mτ k + 1{ i∈Γ(Y k+1 ) } . (15) τk+1 ` If Xτ k0 = 0 for some k0 < +∞ , we set τ = τ 0 , for every ` ≥ k0 i τ 0 := inf{s > 0 : max Xs = 0} = inf{s > 0 : Xs = 0} , (16) 1≤i≤N k and stop the process, i.e., Xt ≡ 0 for t ≥ τ 0 . This way we construct (X·,M·) until time τ (≤ τ 0) for every k ≥ 1 . N Proposition 1. Given a standard Brownian motion W· and the initial configuration X0 ∈ (0, ∞) the process (X·,M·) constructed by this recipe (12)-(15) is the unique, strong solution to (1) with (2), (3) on [0, τ 0] , such that if there is a default, i.e., |Γ(Xt−)| ≥ 1 at time t , then the post-default i i behavior is determined by Xt = Φ (Xt−) for i = 1,...,N . Proof. Because of the similarity of (1) to the particle system (10) discussed in [8], we may adopt the main idea of the proof of their Lemma 3.3. Indeed, at every stopping time τ k , we observe that |Γ(Xτ k−)|(≥ 1) of default events occur, that is, the sample path of the process (X·,M·) has j positive jumps. Because of (11) and the jump sizes of M· with X j j X k (M k − M k ) = 1{j∈Γ(Y k )} = |Γ(Yτ k )| − 1{ i ∈ Γ(Y k ) } , τ τ − τk τk j6=i j6=i the jump sizes of X· in (15) given by
i i i k k 1 1 k Xτ k − Xτ k− = Φ (Yτ k ) − Yτ k = Y τ k · 1 + 1{ i ∈ Γ(Y k ) } − |Γ(Yτ k )| N τk N
i i 1 X j j = X k · M k − M k − M − M ; k = 1, 2,..., τ − τ τ − N τ k τ k j6=i are equal to those in (8) induced by (1). In between the stopping times, the solution to (14) is uniquely determined by the same dynamics as (1) with (2) on a probability space (Ω, F, P) . P1 Note that if N = 1 , then X· = X· with i=1 b(xi, xi) ≡ 0 and hence τ 0 < +∞ a.s. in Lemma 1. In general, the probability that the first passage time τ 0 of zero for the average process is finite depends on the specification of drift function b(·, ·) in (1). For the system with (4) or (6), instead of (1), we may construct the corresponding physical solutions as in Proposition 1. As we have seen in (5), the first passage time of zero for the average process is finite almost surely in the system (4).
2.2 Mean-Field Approximation 2.2.1 Informal Derivation of Mean-Field Limits Let us discuss a mean-field approximation of McKean-Vlasov type for the system (1) with (2)- (3). In this section let us assume b(x, m) = −a(x−m) , x, m ∈ [0, ∞) for some a > 0 , and assume further that the empirical distribution N N 1 X Ft (·) := δ i ; t ≥ 0 N Xt i=1
7 converges weakly to a law of process {Xt, t ≥ 0} described by
Z t Z t Xt = X0 − a (Xs − E[Xt])ds + Wt + E[Xs−]d(Ms − E[Ms]) ; t ≥ 0 , (17) 0 0
P∞ k k−1 where W· is the standard Brownian motion, Mt := k=1 1{τ k≤t} , τ := inf{s > τ : Xt− ≤ 0} , k ≥ 1 , τ 0 = 0 . Then taking expectations of both sides of (17), we obtain
E[Xt] = E[X0] =: x0 ; t ≥ 0 .
When E[X0] =: x0 for some x0 > 0 , substituting this back into (17), we obtain Z t Xt = X0 − a (Xs − x0)ds + Wt + x0(Mt − E[Mt]) ; t ≥ 0 . (18) 0
Transforming the state space from [0, ∞) to (−∞, 1] by Xbt := (x0 − Xt) / x0 , we see Z t Xbt = − aXbsds + Wct − Mct + E[Mct]; t ≥ 0 , (19) 0 where Wc· = W· / x0 , Mc· = M· . This nonlinear McKean-Vlasov-type equation can be seen as the mean field limit of the transformed process in (9). This transformed process Xb· is similar to the nonlinear McKean-Vlasov-type stochastic dif- ferential equation
Z t Xet = Xe0 + b(Xes)ds + Wft − Mft + αE[Mft]; t ≥ 0 , (20) 0 studied in [8, 9]. Here X0 < 1 , α ∈ (0, 1) , b : (−∞, 1] → R is assumed to be Lipschitz continuous P∞ with at most linear growth. W· is the standard Brownian motion, M· = 1 k with f f k=1 {τe ≤·} k k−1 0 τe := inf{s > τ : Xes− ≥ 1} , k ≥ 1 , τe = 0 . When we specify Xe0 = 0 , b(x) = −ax , x ∈ R+ , and α = 1 , the solution (Xb·, Mc·) to (20) reduces to the solution (Xe·, Mf·) to (19), however, the previous study of (20) does not guarantee the uniqueness of solution to (20) in the case α = 1 .
2.2.2 Uniqueness of the Mean Field Limit
Following [9], we may reformulate the solution (Xb·, Mc·) to (19) by Z t Zbt := Xbt + Mct = −a (Zbs − Mcs)ds + Wct + E[Mct] , (21) 0
+ Mct = b sup (Zbs) c ; t ≥ 0 . (22) 0≤s≤t
Here b·c is the integer part. Given a candidate solution et for E[Mct] , t ≥ 0 , we shall consider Z t e e e e e + Zbt = −a (Zbs − Mcs)ds + Wct + et , Mct = b sup (Zbs ) c ; t ≥ 0 , (23) 0 0≤s≤t
8 e e where the superscripts e of Zb· and Mc· represent the dependence on e· . Then uniqueness of ∗ ∗ the solution to (19) is reduced to uniqueness of the fixed point e· = M·(e ) of the map M : C(R+, R+) → C(R+, R+) defined by
e + e Mt(e) := E b sup (Zbs ) c = E[Mct ]; t ≥ 0 . (24) 0≤s≤t
This can be verified by the observation
Z t Z t e∗ e∗ e∗ e∗ ∗ e∗ Xbt = Zbt − Mct = −a (Zbs − Mcs )ds + Wct + et = −a Xbsds + Wct + E[Mct ] 0 0 for every t ≥ 0 .
2.2.3 Numerical Approximation of Fixed Point
1 2 1 2 The map e → M(e) in (24) is monotone, in the sense that if e· , e· ∈ C(R+, R+) with et ≤ et for every t ≥ 0 , then 1 2 Mt(e ) ≤ Mt(e ); t ≥ 0 . (25) (0) With this idea, let us consider the following numerical approximation. Start with e· ≡ 0 and define recursively (n+1) (n) et = Mt(e ); n ≥ 0 , t ≥ 0 . (26) (n+1) By the definition of the map, t → et is strictly increasing for n ≥ 0 . Then by the monotonicity (1) (0) of the map M and et ≥ et ≡ 0 , t ≥ 0 , we have
(0) (1) e· ≤ e· ≤ ..., (27) and hence, we conjecture that if the limit
(n) e∗ := lim e (28) t n→∞ t
∗ exists and is finite for every t ≥ 0 , then e· serves as the fixed point of the map M , that is,
∗ ∗ Mt(e ) = et ; t ≥ 0 .
To discuss the convergence (28), we shall consider the sup norm kekT := sup0≤s≤T |e(s)| for every T > 0 and evaluate
(n+2) (n+1) (n+1) (n) ke − e kT = kM·(e ) − Mt(e )kT e(n+1) + e(n) + (29) = sup E b sup (Zbs ) c − b sup (Zbs ) c 0≤t≤T 0≤s≤t 0≤s≤t
(n+1) (n) in terms of ke − e kT . We define {x} := x − bxc , the non-integer part of x ∈ [0, ∞) .
9 2.2.4 Case a = 0 of No Drifts e In the special case when a ≡ 0 , we have Zb· = Wc· + e· in (23). In this case we may evaluate (29). First observe the identity
bxc − byc = bx − yc + 1{{x}<{y}} ; 0 ≤ y ≤ x < ∞ on the integer part b·c and non-integer part { · } . Applying this identity inside the expectation in (29), we obtain for every n ≥ 0
(n+2) (n+1) (n+1) (n) ke − e kT − bke − e kT c (n+1) + (n) + (n+1) (n) = sup E b sup (Wcs + es ) c − b sup (Wcs + es ) c − bke − e kT c 0≤t≤T 0≤s≤t 0≤s≤t (n+1) + (n) + (n+1) (n) = sup E[b sup (Wcs + es ) − sup (Wcs + es ) c] − bke − e kT c 0≤t≤T 0≤s≤t 0≤s≤t (30) (n+1) + (n) + + P({ sup (Wcs + es ) } < { sup (Wcs + es ) }) 0≤s≤t 0≤s≤t (n+1) + (n) + ≤ sup P { sup (Wcs + es ) } < { sup (Wcs + es ) } . 0≤t≤T 0≤s≤t 0≤s≤t In the last inequality of (30) we used
(n+1) + (n) + (n+1) (n) | sup (Wcs + es ) − sup (Wcs + es ) | ≤ ke − e kT ; 0 ≤ t ≤ T. (31) 0≤s≤t 0≤s≤t By (30) we have an easy upper bound
(n+2) (n+1) (n+1) (n) ke − e kT ≤ bke − e kT c + 1 ; n ≥ 0 , and hence n (n) (n) X (k) (k−1) (1) (1) eT = ke kT ≤ ke − e kT ≤ n(bke kT c + 1) + {ke kT } < +∞ ; n ≥ 0 . k=1
(1) (0) Here e· = M·(e ) = M·(0) is evaluated as ∞ ∞ √ (1) + X X et = E[b sup (Wcs) c] = P( sup Ws ≥ kx0) = P(|W1| ≥ kx0/ t ) 0≤s≤t 0≤s≤t k=1 k=1 √ ∞ ∞ −u2/2 ∞ −k2x2/(2t) X Z 2 e X 2 e 0 t = √ √ du ≤ √ k=1 kx0/ t 2π k=1 x0k 2π √ ∞ −k2x2/(2t) s 2 e 0 t t 2 t X −x0/(2t) ≤ √ = 2 Θ3(0, e ) − 1) ≤ ; t ≥ 0 , 2πx x0 k=1 x0 2π 0 where Θ3(·, ·) is the Jacobi elliptic theta function. The first inequality follows from the tail (1) estimate of the Gaussian probability. Thus the curve t → et is bounded by the line with the slope (1) (1) (1) 1 / x0 and zero intercept. The first and second derivatives e˙t , e¨t of t → et are given by
(1) ∞ −k2x2/(2t) (1) de X kx0e 0 e˙ := t = √ ≥ 0 , t d t 3 k=1 2πt
10 2 (1) ∞ −k2x2/(2t) 2 2 (1) d e X kx0e 0 3 k x e¨ := t = √ − + 0 , t d t2 3 2t 2t2 k=1 2πt (1) (1) (1) (1) for t ≥ 0 with e˙0+ = 0 =e ¨0+ and limt→∞ e˙t = 0 = limt→∞ e¨t . Then it is natural to consider the family of functions n t o L := e ∈ C([0, ∞), [0, ∞)) : e0 = 0 , et ≤ `(t) := ; t ≥ 0 . (32) x0
Proposition 2. Assume x0 ≥ 1 and a = 0 . For every e ∈ L in (32) we have M(e) ∈ L . In (n) particular, e· defined in (26) belongs to L for every n ≥ 0 .
Proof. For every e ∈ L we have et ≤ `(t) = t/x0 , t ≥ 0 and hence if x0 ≥ 1 , then
∞ ∞ + X + X + 0 ≤ Mt(e) = E b sup (Wcs +es) c = P sup (Wcs +es) ≥ k ≤ P sup (Ws +s) ≥ kx0 0≤s≤t 0≤s≤t 0≤s≤t k=1 k=1
∞ r r 1 X kx0 t kx0 t t = Erfc √ − + e2x0k Erfc √ + ≤ (33) 2 2 2 x0 k=1 2t 2t for every t ≥ 0 . The last inequality in (33) may be directly verified in some numerical approximation of the infinite series by the corresponding finite sum. See Appendix C for the formal proof of the last inequality in (33) by the renewal theory. If x0 < 1 , then the last inequality (33) does not necessarily hold for some small t ≥ 0 . Thus we obtain the claim.
The differentiability of t → Mt(e) may be shown as in Proposition 3.1 of [8].
If x > y but {x} < {y} for some x, y ∈ R+ , then {x − y} = 1 + {x} − {y} ≥ {x} and hence, bxc ≤ x = bxc + {x} ≤ bxc + {x − y} . This observation with (31) implies that if (n+1) (n) (n+1) (n) (n+1) (n) ke − e kT < 1 , i.e., ke − e kT = {ke − e kT } , then
n (n+1) + (n) + o { sup (Wcs + es ) } < { sup (Wcs + es ) } 0≤s≤t 0≤s≤t
∞ [ n (n+1) + (n+1) (n) o ⊆ k ≤ sup (Wcs + es ) < k + {ke − e kT } 0≤s≤t k=1 and hence,
(n+1) + (n) + sup P { sup (Wcs + es ) } < { sup (Wcs + es ) } 0≤t≤T 0≤s≤t 0≤s≤t ∞ (34) X (n+1) + (n+1) (n) ≤ sup P sup (Wcs + es ) ∈ (k, k + {ke − e kT }) . 0≤s≤t 0≤t≤T k=1
(n+1) (n) Now let us write ε := {ke − e kT } ∈ (0, 1) . Combining (34) with the inequality
∞ ∞ + X (n+1) + X s P sup (Wcs + es ) ∈ (k, k + ε) ≤ P sup Wcs + ∈ (k, k + ε) , (35) 0≤s≤t 0≤s≤t x0 k=1 k=1
11 we may find δ0,T,x0 ∈ (0, 1) such that
(n+1) + (n) + sup P { sup (Wcs + es ) } < { sup (Wcs + es ) } 0≤t≤T 0≤s≤t 0≤s≤t ∞ X s + ≤ sup P sup Wcs + ∈ (k, k + ε) 0≤s≤t x0 0≤t≤T k=1 ∞ r r X 1h x0k T x0k T (36) = Erfc √ − + e2x0kErfc √ + 2 2 2 k=1 2T 2T r r x0(k + ε) T x0(k + ε) T i − Erfc √ − − e2x0(k+ε)Erfc √ + 2T 2 2T 2 (n+1) (n) ≤ δ0,T,x0 · ε = δ0,T,x0 · {ke − e kT }
(n+1) (n) for every n ≥ 0 . Note that lim 2 δ0,T,x = 1 . Thus if ke − e kT < 1 , then combining T/x0→∞ 0 (30) with (36), we obtain
(n+2) (n+1) (n+1) (n) ke − e kT ≤ δ0,T,x0 ke − e kT ; n ≥ 0 . (37)
(1) (1) (0) (1) Since we have et ≤ t / x0 by Lemma 2 and ke − e kt = et < 1 for 0 ≤ t < x0 , we have the following conjecture. Conjecture : Assume that {e(n)} is generated by the recipe in (26). For every T there exists δ ∈ (0, 1) such that
(n+1) + (n) + (n+1) (n) sup P { sup (Wcs + es ) } < { sup (Wcs + es ) } ≤ δ {ke − e kT } ; n ≥ 0 . (38) 0≤t≤T 0≤s≤t 0≤s≤t
If this conjecture holds, then we see the contraction
(n+2) (n+1) (n+1) (n) (n+1) (n) (n+1) (n) ke −e kT ≤ bke −e kT c+δ {ke −e kT } ≤ 1−(1−δ)c ke −e kT (39)
(n+1) (n) (n+1) (n) conditionally on {ke −e kT } / ke −e kT ≥ c > 0 for some constant c > 0 . If x0 ≥ 1 , ∗ (n) a unique limit et = limn→∞ et , 0 ≤ t < x0 exists and satisfies the fixed point property:
∗ ∗ et = Mt(e ) ; 0 ≤ t < x0 .
i Figure 1 shows the convergence of Picard iteration of the map M·(e ) , i = 1, 2, with initial 0 input e· ≡ 0 in (24), when a = 0 and the initial value x0 is distributed in a stationary distribution from section 2.4.
12 5 4 3 2 1 0
0 1 2 3 4 5
t
i Figure 1 – The iteration of the map Mt(e ) , 0 ≤ t ≤ T , i = 1, 2,..., 21 in (24) is shown under ∗ the stationary initial distribution. A fixed point et , 0 ≤ t ≤ T of the map M·(·) is shown as the maximum curve, when a = 0 .
2.3 Evolutionary FP equation Derivation of the Fokker-Plank equation Let us consider the stochastic integral equation
Z t Z t Z t Xt = X0 + (−a) (Xs − E[Xs])ds + Wt + E[Xs−]dMs − α E[Xs−]dsE[Ms]; t ≥ 0 , (40) 0 0 0
P∞ k k−1 where a ≥ 0 , α ∈ R , Mt := k=1 1{τ k≤t} and τ := inf{s > τ : Xs− ≤ 0} , k ≥ 1 with 0 τ := 0 , and W· is a Brownian motion on a filtered probability space. Assume for a moment that the process {Xt, t ≥ 0} is well defined with a uniquely determined probability distribution on some probability space (Ω, F, P) . Here Mt is cumulative number of default events {0 ≤ s ≤ t : Xs = 0} until time t ≥ 0 . We take t → Mt as a càdlàg process, i.e., right continuous with left limits. Assuming t → E[Xt] and t → E[Mt] =: et are smooth, let us introduce its expectation xt := E[Xt] and derivative e˙t = dE[Mt]/dt , t ≥ 0 . Thus we may rewrite the dynamics Z t Z t Z t Xt = X0 + (−a) (Xs − xt)ds + Wt + xsdMs − α xse˙sds ; t ≥ 0 , (41) 0 0 0 where xt = x0 · exp((1 − α)et) , t ≥ 0 . The probability density function p(t, x)dx = P(Xt ∈ dx) of Xt solves the Fokker-Plank equa- tion 1 2 ∂tp(t, x) + ∂x[(−a(x − xt) − αxte˙t) p(t, x)] − ∂ p(t, x) =e ˙t δx (dx) (42) 2 xx t for t > 0 , x > 0 , where δx(dx) is a Dirac measure at x . For the boundary condition we assume that lim p(t, x) = 0, lim p(t, x) = 0 , lim ∂xp(t, x) = 0 , (43) x↓0 x→+∞ x→∞
lim p(t, x) = P(X0 ∈ dx)/dx , (44) t↓0 and Z ∞ 1 xt = xp(t, x)dx , e˙t = ∂xp(t, 0) ; t ≥ 0 . (45) 0 2
13 See Appendix A for more details on the derivation of this equation. We note that if the mean-field term xt is forced to be a constant (say x0 > 0 ), then the corresponding PDE is
1 2 ∂tp(t, x) + ∂x[(−a(x − x0) − αx0e˙t) p(t, x)] − ∂ p(t, x) =e ˙t δx (dx) 2 xx 0 for (t, x) ∈ (0, ∞)×(0, ∞) , and then after a change of variables y = (x0−x)/x0 , pb(t, (x0−x)/x0) := p(t, x) , we obtain another Fokker-Planck equation
1 2 ∂tpb(t, y) + ∂y[(−ay + αe˙t)pb(t, y)] − 2 ∂yypb(t, y) =e ˙tδ0(dy) (46) 2x0 for (t, y) ∈ (0, ∞) × (−∞, 1) with the condition corresponding to (43)-(45). Our Fokker-Planck equation (46) is an extension to the Fokker-Planck equation studied in [3]:
1 2 ∂tp(t, y) + ∂y[(−y + αe˙t)p(t, y)] − ∂ p(t, y) =e ˙tδ0(dy) . b b 2 yy b
Indeed, with a = 1 and x0 = 1 , (46) reduces to the study in [3].
Notion of solution and blow-up phenomenon We borrow the following notion of solution from [3]. ∞ + 1 Definition 3. We say that a pair of nonnegative functions (p, e˙) with p ∈ L (R ; L+(0, +∞)) and 1 + e˙ ∈ Lloc,+(R ) is a weak solution of (42)–(45) with initial condition p0(·) := p(0, x), if for any test ∞ 2 ∞ function (t, x) 7→ φ(t, x), φ ∈ C ([0, +∞) × [0,T ]) such that ∂xxφ, x∂xφ ∈ L ([0, +∞) × (0,T )), and we have Z T Z +∞ 1 2 p(t, x) −∂tφ(t, x) − ∂xφ(t, x)(−a(x − x0) − x0e˙t) − ∂xxφ(t, x) dx dt 0 0 2 Z T Z +∞ Z +∞ = e˙t [φ(x0) − φ(0)] dt + p0(x)φ(0, x) dx − p(T, x)φ(T, x) dx. 0 0 0 By choosing test functions of the form φ(t, x) = ψ(t)φ(x) and differentiating with respect to time variables, this definition is equivalent to having the following equation satisfied for every ∞ ∞ φ ∈ C ([0, +∞)) with x∂xφ ∈ L ((0, +∞)), d Z +∞ φ(x)p(t, x)dx dt 0 (47) Z +∞ 1 2 = ∂xφ(x)(−a(x − x0) − x0e˙t) + ∂xxφ(x) p(t, x) dx +e ˙t [φ(x0) − φ(0)] . 0 2 The complete analysis of the existence and uniqueness of the weak solution is beyond the scope of our current study. In the following, we shall point out that the weak solution does not exist globally in time due to the blow-up phenomena, if the initial distribution p0(·) = p(0, ·) concentrates near the origin.
Theorem 4 (Blow-up phenomenon). Fix a ∈ R and x0 > 0 . If there exists µ > max(2ax0, 1) such that the initial condition p0(·) satisfies
Z ∞ −µx0 −µx 1 − e e p0(x)dx ≥ , (48) 0 µx0 then there are no global-in-time weak solutions to (42)–(45).
14 Proof. The proof follows the lines of [3, Theorem 2.2], adapted to our setting. Let us assume there exists a global-in-time weak solution, in the sense of Definition 3 with a test function φ(t, x) = −µx R ∞ φ(x) = e . Let us define the Laplace transform Mµ(t) = 0 φ(x)p(t, x)dx of p(t, x) . Notice that x µe˙ ≥ 0 for all t ≥ 0 and M (0) ≥ λ by (48) with λ := φ(0)−φ(x0) > 0. By (47), we have 0 t µ x0 µ Z +∞ d 1 2 Mµ(t) = −µφ(x)(−a(x − x0) − x0e˙t) + µ φ(x) p(t, x) dx − λµe˙t dt 0 2 1 ≥ µ x0e˙t + µ − ax0 Mµ(t) − λµe˙t (49) 2 λ ≥ x0µe˙t Mµ(t) − , (50) x0 where we used the fact that x ≥ 0 and µ ≥ 2ax0. Hence, by the Gronwall inequality, (50) implies λ Mµ(t) ≥ , ∀t ≥ 0. (51) x0 Going back to (49), we obtain
d 1 Mµ(t) ≥ µ µ − ax0 Mµ(t), dt 2 which implies, again by the Gronwall inequality,
1 1 µ[ µ−ax0]t µ[ µ−ax0]t λ Mµ(t) ≥ e 2 Mµ(0) ≥ e 2 · . x0
1 Since 2 µ − ax0 > 0, the right hand side grows to +∞ as t → +∞. On the other hand, since φ(x) = e−µx ≤ 1 and p is a probability density, Z ∞ Z ∞ Mµ(t) = φ(x)p(t, x)dx ≤ p(t, x)dx ≤ 1, 0 0 which yields a contradiction when t is large enough.
For example, if the initial condition p0(·) takes a form of triangular distribution supported by 2 2 an open interval (0, 2c) with p0(x) = 1{0 2 2 Z ∞ x 2 −2x −x0 −µx (e 0 − 1) e 0 1 − e e p0(x)dx = 4 > . 0 x0 x0 The probability of this triangular distribution p0(·) is concentrated near the origin, and by Theorem 4, there is no global-in-time weak solution to (42)–(45). The existence of steady states and the convergence to such stationary distributions have been addressed respectively in [3, Theorems 3.1 and 4.1]. However, our dynamics does not fit in the assumptions made for the aforementioned results (we are in the regime where, using the notations of [3], b = VF − VR = 1 and the function h depends on N). For this reason, we address the question of steady states in the next section by directly finding an explicit solution. 15 2.4 Explicit solution for the stationary FP equation Let us look for a stationary solution to the Fokker-Planck equation (42)–(45). In other words, we look for a function p :(−∞, 1) → R such that, at least in a weak sense, d 1 00 [(−a(x − x0) − x0e0) p(x)] − p (x) = e0 δx (dx) (52) dx 2 0 for t > 0 , x > 0 . For the boundary condition we assume that lim p(x) = 0, lim x p(x) = 0 , lim p0(x) = 0 , (53) x↓0 x→+∞ x→∞ and 1 0 e0 = p (0) . (54) 2 We show that there exists an explicit solution to the stationary Fokker-Planck equation. Theorem 5. (i) When a > 0, the following is a solution to (52)–(54) : for all x ∈ [0, +∞), min(x,x0) ! Z 2 2 ay +2x0(e0−a)y −ax −2x0(e0−a)x p(x) = 2e0 e dy e , (55) 0 where e0 is uniquely determined by : q 2 Z e0x0 2 e0 a y 2 q √ e N (−y)dy = 1 , (56) a 2 a e0− 2a x0 z −x2 R 2 where N (z) = −∞ e dx, and we have an upper bound estimate 2a 1 2 2 0 < e0 < max 2 , 2 (e (1 + 2ax0) − 1) . (57) log(2ax0) 2x0 (ii) When a = 0, a solution is given by : ! Z min(x,x0) 2x0e0y −2x0e0x p(x) = 2e0 e dy e , (58) 0 with 1 e0 = 2 . (59) x0 The proof is given in Appendix B. The above result allows us to study how the default rate depends on the parameters of the model. As shown on Figure 2, e0 is decreasing with respect to a (the mean-reverting term has a stronger effect and stabilizes the system) and it is also decreasing with respect to x0 (banks start further away from 0). 16 Figure 2 – e0 as a function of a and x0. 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 density density 0.4 0.4 density 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 x x x (a) (b) (c) Figure 3 – Density for a = 0.01125,X0 = 2.0 : analytical solution (55) (left) and its approximation using 106 Monte-Carlo samples following the dynamics (1) (middle) and (60) (right), for t = 100. In red (dashed line) are plotted respectively x0, and the empirical averages Xt and X t. 17 1.0 1.0 1.0 0.8 0.8 0.8 0.6 0.6 0.6 density density 0.4 0.4 density 0.4 0.2 0.2 0.2 0.0 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 x x x (a) (b) (c) Figure 4 – Density for a = 2.0,X0 = 2.0 : analytical solution (55) (left) and its approximation using 106 Monte-Carlo samples following the dynamics (1) (middle) and (60) (right), for t = 100. In red (dashed line) are plotted respectively x0, and the empirical averages Xt and X t. 2.5 Numerical approximation of the solution As shown in Figures 3a and 4a, a larger value of α induces more concentration around x0 in the density (55). The numerical results also show that, when t and N are large, this density is well approximated by the empirical distribution of Monte-Carlo simulation of trajectories according to either (1) or an i i N-particle approximation of the mean field dynamics (18) with states X. , M., namely: Z t Z t i i i i i 1 X j X = X + b(X , x0)ds + W + x0 dM − dM ; t ≥ 0 , t 0 s t s N s 0 0 j6=i ∞ (60) i X i n i i x0 X j j o Mt := 1{τ i ≤t} , τk := inf s > τk−1 : Xs− − (Ms − Ms−) ≤ 0 ; k ∈ N , k N k=1 j6=i Conversely, it also means that the long time behavior of both systems of SDEs is well approximated by the mean field dynamics (55) in the stationary regime. 1 PN i Figure 5 displays the evolution of the average liquidity amounts, X and X = N i=1 X . Although the systems are expected to have the same mean-field limit in which the mean is constant, here the values fluctuate around x0 = 2.0 because there is only a finite number of agents. Note that the fluctuations are the smallest when using a large a (hence a strong mean-reverting effect) and a fixed level for births (namely, x0). P i For the dynamics (1) (respectively (60)), the default rate is given by d i M /dt (resp. P i P i P i d i M /dt), whereas the cumulative number of defaults is simply i M (resp. i M ). Their evolutions are displayed on Figure 6. At time t = 0, the default rate is 0 because the initial distribution is X0 = x0 > 0 a.s.; then, it converges towards a stationary value. 18 0 00 0 00 X X X X 0 00 0 00 X X0 X X0 0 00 0 00 0 000 0 000 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 00 0 0 0 0 0 00 0 0 0 0 0 00 Figure 5 – Evolution of the empirical average of the N-dimensional process X, for N = 106 Monte-Carlo samples following the dynamics (1) (in green, dashed curve) and (60) (in blue), with a = 0.01125 (left) and a = 2.0 (right). In both cases, the initial distribution is concentrated at X0 = 2.0 (dashed line, in red). 0.025 2.0 1e7 0.020 1.5 0.015 1.0 0.010 default rate default number of defaults of number 0.5 0.005 i i d( iM /dt (renew at X) ΣiM (renew at X) i i d( iM /dt (renew at X ) ΣiM (renew at X ) 0.000 0.0 0 20 40 60 80 100 0 20 40 60 80 100 time time Figure 6 – Evolutions of the default rate (left) and of the cumulative number of defaults (right), 6 for the dynamics (1) (resp. (60)), with N = 10 Monte-Carlo samples, a = 0.01125 and X0 = 2.0. The curves correspond to the dynamics (1) (in green, dashed curve) and (60) (in blue). 3 Mean Field Game Model We now turn our attention to the situation where the agents can control their dynamics and try to optimize a certain criterion. Based on the previous section, we focus directly on the macroscopic description. We want to study Nash equilibria and, to this end, we use the framework of mean field games. The players interact through two mean field terms: the average wealth and the rate of defaults. 19 3.1 Formulation of the problem In order to describe the mean field game problem, we will use the following notations. Let P be the family of probability densities over R+ = [0, +∞) with a right-hand derivative at x = 0. For a probability density m ∈ P, we define its first moment Z +∞ m(m) = xm(x)dx , (61) 0 and, σ2 d e(m) = m(0) . (62) 2 dx For a random element Z, we denote by L(Z) the law of Z. In particular, we consider a flow of density function mt of the marginal law at time t ≥ 0 for a generic stochastic process. Here and thereafter, to allow more flexibility in the numerical investigation, we do not assume anymore that the diffusion coefficient σ is necessarily 1. Controlled dynamics. For each flow of densities m = (mt)t≥0 and each control ξ = (ξt)t≥0 with mt : R+ → R and ξt : R+ → R for each t ≥ 0, we consider the following two dynamics m,ξ ˜ ξ (Xt )t≥0 and (Xt )t≥0 : Z t Z t m,ξ m,ξ m,ξ Xt = X0 + b Xs , ms, ξs(Xs ) dt + σdWs ; t ≥ 0 , (63) 0 0 Z t Z t Z t ˜ ξ ˜ ξ ˜ ξ ˜ ξ ˜ ξ ˜ ξ Xt = X0 + b Xs , L(Xs ), ξs(Xs ) dt + m L(Xs ) dMs + σdWs ; t ≥ 0 , (64) 0 0 0 where the initial value X0 is a positive random variable, W is a standard Brownian motion, and ˜ ξ k,ξ Mt is the cumulative number of defaults on or before time t which occur at τ , k ≥ 1, i.e., ∞ ˜ ξ X k,ξ k−1,ξ ˜ ξ 0,ξ Mt = 1{τ˜k,ξ≤t} , τ˜ = inf{s > τ˜ : Xt− ≤ 0} , k ≥ 1 , τ˜ = 0 . k=1 With (61)-(62) the drift functional b is defined by b(x, m, ξ) = ξ + a(m(m) − x) − γe(m)m(m) , where γ ∈ (0, 1] is a constant. The case ξ = 0 and γ = 1 corresponds to the uncontrolled dynamics studied in the previous section. ξ Furthermore, for the expression (64) to make sense, we assume that L(X˜s ) has a density with respect to Lebesgue measure, and we identify it with this density. m,ξ ˜ ξ m,ξ The interpretation of the above two processes (Xt )t≥0 and (Xt )t≥0 is the following: X describes the state of an infinitesimal player using control ξ when the flow of densities of the rest of the population is given by m, while L(X˜ ξ) describes the state of the population when each (infinitesimal) player uses the control ξ. Here, we assume that each bank controls its rate of borrowing or lending to a central bank through the (feedback) control rate ξ : R+ × R+ → R as a function of time and state. Notice that the stochastic differential equation of X˜ ξ is of the non-linear, McKean-Vlasov type, m,ξ ˜ m,ξ whereas the dynamics of X is not, since m is fixed. Moreover, due to the control ξ, E[Xt ] is not necessarily constant. 20 Objective function. Following Carmona et al. in [6] for a systemic risk model, we consider a running cost f defined by 1 f(x, m, ξ) = ξ2 − qξ(m(m) − x) + (m(m) − x)2 . (65) 2 2 where q > 0 and > 0 are parameters of the problem. Here, the parameter q is interpreted as the incentive to borrowing or lending: The bank will borrow (ξ > 0), if x < m(m) and lend (ξ < 0), if x < m(m). Equivalently, after dividing by q, this parameter can be seen as a control by the financial regulator of the cost of borrowing or lending (q large meaning low fees). Furthermore, the quadratic term (m(m) − x)2 in the running cost penalizes deviations from the average. We assume that q2 ≤ , so that for a fixed m, (x, ξ) 7→ f(x, m, ξ) is convex. Notice, as a special case, 2 1 2 that if q = , then it is simply f(x, m, ξ) = 2 (ξ − q(m(m) − x)) . Let r > 0 be an instantaneous discounting rate parameter and m0 be an initial probability density. We introduce the following objective function (or cost functional) for each infinitesimal player m,ξ with state process (Xt )t≥0 in (63). For a density flow m = (mt)t≥0 and an admissible control ξ = (ξt)t≥0, we let the objective cost function be : "Z τ m,ξ # m −rs m,ξ m,ξ J (ξ) = E e f(Xs , ms, ξs(Xs ))ds , (66) 0 where Xm,ξ is given by (63), and τ m,ξ denotes the first time Xm,ξ hits 0 : m,ξ m,ξ τ = inf{s > 0 : Xs− ≤ 0} . (67) This is the cost that a representative (and infinitesimal) player with dynamics (63) tries to minimize, when the dynamics of the population is described by m. Mean field game. The mean field game we consider is defined as the problem of finding (m, ˆ ξˆ) such that the following two conditions are satisfied: 1. ξˆ minimizes J mˆ ; ˜ ξˆ 2. mˆ t = L(Xt ), for all t ≥ 0 . The first condition means that ξˆ is the best response control of an infinitesimal player facing the population whose distribution is given by mˆ . The second condition ensures consistency of the best response and the population’s behavior. Overall, the mean field game can be construed as a fixed point problem. We will also be interested in a similar game but with finite time horizon. Although this trunca- tion might be a bit artificial, it is more convenient in order to solve numerically the problem. We thus fix a time horizon 0 < T < +∞ and consider, instead of (66), the following objective function "Z τ m,ξ∧T # m −rs m,ξ JT (ξ) = E e f(Xs , ms, ξs)ds . (68) 0 m m We expect that for each ξ = (ξt)t≥0, JT (ξ) → J (ξ) as T → ∞. As above, the solution to the mean field game on a finite time horizon is defined as a fixed point. We leave for future work the challenging questions of existence and uniqueness of a solution in the general setting. We will however provide below an explicit solution in a special case and then we compute an (approximate) solution for a discrete version of the problem in the general case. 21 3.2 PDE system for the Mean Field Game We now want to characterize the MFG solutions in the form of a PDE system. Following the approach of Lasry and Lions [22], we obtain the system consisting of a forward Fokker-Plank (FP) equation for the evolution of the density of the population, and a backward Hamilton-Jacobi- Bellman (HJB) equation for the evolution of the value function of an infinitesimal player. We define the Hamiltonian: for x ∈ R+, m ∈ P and p ∈ R, H(x, m, p) = − inf {b(x, m, ξ)p + f(x, m, ξ)} (69) ξ The infimum above is achieved by ξ = Ξ(x, m, p) = −p − q(x − m(m)). Hence 1 1 H(x, m, p) = p2 − (q + a)(m(m) − x) − γe(m)m(m)p + (q2 − )(m(m) − x)2 . 2 2 To alleviate the notations, let us also introduce: ϕ(x, m) = (q + a)(m(m) − x) − γe(m)m(m) , 1 ψ(x, m) = ϕ(x, m)2 − (q2 − )(m(m) − x)2 , 2 where x ∈ R+ and m is a density over R+. Then, the Hamiltonian can be written as 1 H(x, m, p) = [p − ϕ(x, m)]2 − ψ(x, m) . (70) 2 It can be shown that if ξˆ = (ξˆt)t≥0 is an optimal feedback control, then it must satisfy, for all (t, x), ξˆ(t, x) = Ξ(x, mˆ (t, ·), ∂xuˆ(t, x)) = −∂xuˆ(t, x), (71) where (ˆu, mˆ ) solve the following HJB-FP PDE system : for (t, x) ∈ (0, +∞) × (0, +∞), σ2 ru(t, x) − ∂tu(t, x) − ∂xxu(t, x) + H(x, m(t, ·), ∂xu(t, x)) = 0, (72a) 2 2 σ ∂tm(t, x) − ∂xxm(t, x) − ∂x Hp(x, m(t, ·), ∂xu(t, x))m(t, x) = e(m(t, ·))δm(m(t,·))(x), (72b) 2 with the boundary conditions : for all t ∈ (0, +∞), u(t, 0) = 0 and m(t, 0) = 0, and the initial and final conditions : for all x ∈ [0, +∞), m(0, x) = m0(x) u(T, x) = 0 . (73) 3.3 Explicit solution for a stationary Mean Field Game In this section, we focus on a stationary regime. Building on our explicit solution for the dynamics (see Theorem 5), we provide an example of stationary mean field game with an explicit solution. This is useful for instance as a benchmark in order to test numerical methods. To this end, we slightly modify the problem and add a non-trivial boundary condition on u at x = 0, which can be 22 interpreted as an exit cost (or benefit) when the bank defaults. Instead of (68), we consider the objective function "Z τ m,ξ∧T # m −rs m,ξ −rτ m,ξ m,ξ JT (ξ) = E e f(Xs , ms, ξs)ds + e 1{τ m,ξ<+∞}Γ(m(τ , ·)) , (74) 0 m,ξ for some function Γ : [0, ∞) × P → R to be chosen below. Here, τ is the exit time defined in (67). In this case, the boundary condition for u at x = 0 becomes: for all t ∈ (0,T ) u(t, 0) = Γ(m(t, ·)). We then look for a stationary solution. This leads us to consider the PDE system for the unknowns (u, m), functions of x only : for x ∈ (0, +∞), σ2 ru(x) − u00(x) + H(x, m, u0(x)) = 0, (75a) 2 2 σ 00 0 − m (x) − ∂x Hp(x, m, u (x))m(x) = e(m)δm(m)(x), (75b) 2 with the boundary conditions : u(0) = Γ(m) and m(0) = 0 . Let us look for u in the form: 1 u(x) = A(x − m(m))2 + B(x − m(m)) + C (76) 2 where A, B, C are three real numbers to be determined. With this ansatz, we have 1 1 H(x, m, u0(x)) = q2 − + (q + a)A + A2 (x − m(m))2 2 2 + [(q + a)B + AB + Aγe(m)m(m)] (x − m(m)) 1 + B2 + Bγe(m)m(m) , 2 so, rewriting the HJB equation as a polynomial expression in (x − m(m)) and identifying each coefficient to 0 yields √ −(r + 2(q + a)) + ∆ 2 2 A = , ∆ = [r + 2(q + a)] − 4(q − ), (77a) 2 −A B = γe(m)m(m) (77b) q + a + A + r 2 1 σ 1 2 C = A − B − Bγe(m)m(m) (77c) r 2 2 1 Γ(m) = Am(m)2 − Bm(m) + C. (77d) 2 The last equality above comes from the boundary condition for u at x = 0. It can be ensured by suitably choosing Γ, provided we first find A, B, C satisfying the three first equations. Assume that γ satisfies −B − γ = −1 (78) 23 i.e., γ = 1 − A/(q + a + A + r). Then (75b) rewrites σ2 h i − m00(x)−(A+q +a)m(x)− (A+q +a)(x−m(m))+e(m)m(m) m0(x) = e(m)δ (x) , (79) 2 m(m) which is known to have a closed form solution from Theorem 5 on the uncontrolled stationary FP equation. Note that, due to (78), γ can not be 1 unless B = A = 0, which leads to an explicit but trivial solution in the sense that the optimal control is simply 0 in this case. However, for γ ∈ (0, 1), we obtain a non-trivial explicit stationary solution for which the equilibrium stationary control is given according to (71) by ξˆ(x) = −∂xu(x) = −A(x − m(m)) − B. 3.4 Numerical approximation scheme for the general Mean Field Game We will use finite differences to discretize the HJB-FP PDE system (72a)–(72b), adapting to our setting the scheme introduced in [1]. The main difference is that, in the FP equation, we need to deal with the defaults and with the creation of new banks at a value which is not known in advance. For numerical purposes, it will be interesting to consider the PDE system on a truncated domain. Let L > 0 be a given constant, and let us denote by D = (0,L) the spatial domain and by QT = (0,T ) × D the time-space domain. We will consider the PDEs (72a)–(72b) on QT , and impose the following boundary conditions : for all t ∈ (0,T ), u(t, 0) = u(t, L) = 0 , ∂xu(t, L) = 0 , (80) and m(t, 0) = u(t, L) = 0 , ∂xm(t, L) = 0 , (81) and the initial and final conditions : for all x ∈ D, (73) holds. Discretization. Let NT and Nh be two positive integers corresponding respectively to the number of steps in time and space. We consider (NT + 1) and (Nh + 1) points in time and space respectively. Let ∆t = T/NT and h = L/Nh, and tn = n∆t, xi = i h for (n, i) ∈ {0,...,NT } × {0,...,Nh}. We also introduce an extended index function (which implicitly depend on the spatial grid) defined by: for x ∈ D, ind(x) = max{i : xi ≤ x} . Note that for all x ∈ D, we have x ∈ [xind(ξ), xind(ξ)+1). (k) (k) (k) (k) (N +1)×(N +1) We approximate u and m respectively by vectors U and M ∈ R T h , such (k) (k),n (k) (k),n Nh+1 that u (tn, xi) ≈ Ui and m (tn, xi) ≈ Mi for each (n, i). For M ∈ R , we let X σ2 m(M) = h xiMi , e(M) = (M1 − M0) . 2h i 24 0 0 0 0 0 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t = 0 t = 0 0 0 0 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t = 2 t = 2 0 0 0 00 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 t = 3 t = 3 0 0 0 00 0 000 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 00 0 0 t = 5 t = 5 Figure 7 – Comparison of time dependent MFG solution with quadratic ansatz. Left: Dirichlet conditions at x = 0 and x = L in accordance with the quadratic ansatz. Right: Dirichlet condition 0 at x = 0 and Neumann condition 0 at x = L. Here the time horizon is T = 5.0. 25 We introduce the finite difference operators n 1 n+1 n NT +1 (DtW ) = (W − W ), n ∈ {0,...NT − 1},W ∈ R , ∆t + 1 Nh+1 (D W )i = (Wi+1 − Wi), i ∈ {0,...Nh − 1},W ∈ R , h 1 Nh+1 (∆hW )i = − (2Wi − Wi+1 − Wi−1) , i ∈ {1,...Nh − 1},W ∈ R , h2 + + T Nh+1 [∇hW ]i = (D W )i, (D W )i−1 , i ∈ {0,...Nh − 1},W ∈ R . N +1 Discrete Hamiltonian. We first introduce ϕ˜ and ψ˜, defined for (x, M) ∈ R × R h by: ϕ˜(x, M) = (q + a)(m(M) − x) − e(M)m(M) , 1 ψ˜(x, M) = ϕ˜(x, M)2 − (q2 − )(m(M) − x)2 . 2 We then introduce the following discrete Hamiltonian H˜ , whose definition is based on (70), 1 n −2 +2o H˜ (x, M, p1, p2) = (p1 − ϕ˜(x, M)) + (p2 − ϕ˜(x, M)) − ψ˜(x, M) , (82) 2 N +1 where x ∈ D, p1, p2 ∈ R, M ∈ R h . In particular H˜ has the following properties: 1. Monotonicity: H˜ is nonincreasing in p1 and nondecreasing in p2. 2. Consistency: H˜ (x, M, p, p) = H(x, M, p) (where, in the right-hand side, M is identified with the corresponding piecewise linear function defined by its values on the spatial grid) . 1 3. Differentiability: H˜ is of class C with respect to (x, p1, p2). 4. Convexity: (p1, p2) 7→ H˜ (x, M, p1, p2) is convex. Discrete HJB equation. As in [1], we consider the following discrete version of (72a) 2 rU n − (D U )n − σ (∆ U n) + H˜ (x ,M n+1, [∇ U n] ) = 0 , (83a) i t i 2 h i i h i i ∈ {1,...,Nh − 2} , n ∈ {0,...,NT − 1} , n Ui = 0 , n ∈ {0,...,NT − 1} , i ∈ {0,Nh − 1,Nh} , (83b) NT Ui = 0 , i ∈ {0,...,Nh} . (83c) Discrete FP equation. To define a discretization of the FP equation, we consider the weak form of (72b). It involves, among other terms, for a smooth w ∈ C∞(D × [0,T ]), Z − ∂x Hp(x, m(t, ·), ∂xu(t, x))m(t, x) + e(m(t, ·))δm(m(t,·))(x) w(t, x)dx D Z = Hp(x, m(t, ·), ∂xu(t, x))m(t, x) ∂xw(t, x)dx − e(m(t, ·))w t, m(m(t, ·)) , (84) D where we used integration by parts and the boundary conditions. 26 This leads us to introduce the following two discrete operators. For the first part (see [1] for more details), we introduce 1 Bi(U, M) = MiH˜p (xi,M, [∇ U]i) − Mi−1H˜p (xi−1,M, [∇ U]i−1) h 1 h 1 h ˜ ˜ + Mi+1Hp2 (xi+1,M, [∇hU]i+1) − MiHp2 (xi,M, [∇hU]i) . Notice that, with our definition of H˜ (see (82)), − Ui+1 − Ui H˜p (xi,M, [∇ U]i) = − − ϕ˜(x, M) , 1 h h + Ui+1 − Ui H˜p (xi+1,M, [∇ U]i+1) = − ϕ˜(x, M) , 2 h h and similarly for the other terms. N +1 For the second part of (84), we introduce, for M ∈ R h and µ ∈ D, e(M)(x − µ)/h2 , if i = ind(µ) , i+1 2 βi(M, µ) = e(M)(µ − xi−1)/h , if i = ind(µ) + 1 , 0 , otherwise. Considering a piecewise linear function W defined on D by its values Wi = W (xi), i = 0,...,Nh, at the mesh points, we have N Xh e(M)W (µ) = βi(M, µ)Wi . i=0 Then, for the discrete version of (72b) we consider 2 (D M )n − σ (∆ M n+1) − B (U n,M n+1) − β (M n+1, m(M n+1)) = 0 , (85a) t i 2 h i i i i ∈ {1,...,Nh − 2}, n ∈ {0,...,NT − 1} , n Mi = 0 , n ∈ {1,...,NT } , i ∈ {0,Nh − 1,Nh} , (85b) 0 Mi = m0(xi) , i ∈ {0,...,Nh} . (85c) Remark 6. A direct discretization of (72b) would have to deal with a Dirac mass at the point m(m(t, ·)), which is not necessarily a point of the mesh. One advantage of considering the weak formulation as proposed above is to avoid this issue. Numerical method. We now describe how to compute a solution to the above discrete sys- tem (83a)–(83c) and (85a)–(85c). Note that the discrete FP equation (85a) is non-linear in M (due n+1 n+1 to the term βi(M , m(M ))). Trying to solve this equation using e.g. Newton’s method would involve highly non-sparse matrices (due to the term m(M n+1)). To avoid this issue, we propose to employ the iterative procedure described in Algorithm 1. Notice that (86)–(88) is a modified version of the discrete FP equation (85), in which part of the unknown M is replaced by the estimate M (k) from the previous iteration. At convergence, we have M (k+1) = M (k) and hence (85) is satisfied. In our implementation, instead of fixing the number of iterations, we use as convergence criterion the normalized `2-norms of the difference between two iterates of U and two iterates of M. The discrete HJB equation (89)–(91) is solved by Newton’s method. 27 Algorithm 1: Iterative method for the finite difference system (83) & (85) Data: An initial guess (M,˜ U˜); a number of iterations K. Result: An approximation of (u, m) 1 begin 2 Initialize (M (0),U (0)) ← (M,˜ U˜). 3 for k = 0, 1, 2,...,K − 1 do 4 Compute M (k+1) solving n σ2 n+1 (k),n n+1 (k),n+1 (k),n+1 (DtMi) − 2 (∆hM )i − Bi(U ,M ) − βi(M , m(M )) = 0 , (86) i ∈ {1,...,Nh − 2}, n ∈ {0,...,NT − 1} , n Mi = 0 , n ∈ {1,...,NT } , i ∈ {0,Nh − 1,Nh} , (87) 0 Mi = m0(xi) , i ∈ {0,...,Nh} . (88) 5 Compute U (k+1) solving n n σ2 n ˜ (k+1),n+1 n rUi − (DtUi) − 2 (∆hU )i + H(xi,M , [∇hU ]i) = 0 , (89) i ∈ {1,...,Nh − 2} , n ∈ {0,...,NT − 1} , n Ui = 0 , n ∈ {0,...,NT − 1} , i ∈ {0,Nh − 1,Nh} , (90) NT Ui = 0 , i ∈ {0,...,Nh} . (91) 6 return (M (K),U (K)) 28 3.5 Numerical results We now present numerical results obtained using the numerical method described above. For the results displayed in Figure 8, we fixed L = 10 (so that the space domain is [0, 10]), T = 10, q = 0.1, 2 and = q . We consider as a baseline the setting with a = 0.5, x0 = 2.0, r = 0.5, σ = 1.0. We are particularly interested in the evolution of the default rate e(mt) through time. We note the following behavior: The default rate increases as the interaction strength a or the initial mean x0 increases; it decreases when the discount rate r increases or when the volatility σ decreases. All these variations seem quite natural from the point of view of the interpretation of the model. We note in particular that the solution starting with x0 = 3 is very stable since the mean almost does not change and the default rate is very close to zero. In the other settings, the default rate tends to increase rapidly at the beginning of the time interval before taking a more stable value. This can be explained by the fact that the initial distribution is a truncated Gaussian with value 0 at x = 0, so that the default rate is initially null but the randomness (diffusion term) quickly causes the weakest banks to default. As new banks are re-injected in the system around the mean wealth, the overall default rate finds an (almost) stationary value. However, numerical experiments conducted with x0 much smaller than 1.8 did not converge to a solution. As hinted by Theorem 4, this could be related to the fact that the solution blows up before time T = 10 when the initial distribution is concentrated near x = 0. Figure 9 displays the evolution of m and u in the baseline case mentioned above (namely, a = 0.5, x0 = 2.0, r = 0.5, σ = 1.0). Although m remains concentrated close to its original mean, it seems important from a numerical perspective to consider a large enough spatial domain. Indeed, since we artificially impose a Dirichlet boundary at x = L, choosing L too small would imply that the total mass can not be conserved. 29 3.0 0.7 0.6 2.8 0.5 2.6 0.4 ) m m ( 2.4 0.3 m 2.2 0.2 2.0 0.1 0.0 1.8 0 1 2 3 4 5 6 0 2 4 6 8 10 x time t a=0.5, x0=2.0, r=0.5, σ=1.0 a=0.5, x0=2.0, r=0.2, σ=1.0 a=0.5, x0=2.0, r=0.5, σ=1.0 a=0.5, x0=2.0, r=0.2, σ=1.0 a=0.4, x0=2.0, r=0.5, σ=1.0 a=0.5, x0=2.0, r=0.5, σ=0.8 a=0.4, x0=2.0, r=0.5, σ=1.0 a=0.5, x0=2.0, r=0.5, σ=0.8 a=0.5, x0=1.8, r=0.5, σ=1.0 a=0.5, x0=3.0, r=0.5, σ=1.0 a=0.5, x0=1.8, r=0.5, σ=1.0 a=0.5, x0=3.0, r=0.5, σ=1.0 (a) Final density (b) Mean value 0.025 0.020 0.015 ) m ( e 0.010 0.005 0.000 0 2 4 6 8 10 time t a=0.5, x0=2.0, r=0.5, σ=1.0 a=0.5, x0=2.0, r=0.2, σ=1.0 a=0.4, x0=2.0, r=0.5, σ=1.0 a=0.5, x0=2.0, r=0.5, σ=0.8 a=0.5, x0=1.8, r=0.5, σ=1.0 a=0.5, x0=3.0, r=0.5, σ=1.0 (c) Default rate Figure 8 – Comparison for various settings. Here, we used L = 10, T = 10, q = 0.1, = q2. 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Particle systems with singular interaction through hitting times: application in systemic risk modeling. Ann. Appl. Probab., 29(1):89–129, 2019. [27] R. L. Schilling, R. Song, and Z. Vondraček. Bernstein functions, volume 37 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, second edition, 2012. Theory and applications. A Derivation of the KFP equation We shall derive the KFP for the transition probability p(t, x) of stochastic process Xt , t ≥ 0 represented by Z t Z t Z t Xt = X0 + (−a)(Xs − E[Xs])ds + Wt + E[Xs]dMs − α E[Xs]dsE[Ms] , (92) 0 0 0 where α ≥ 0 . Taking the expectations of both sides, we obtain a linear integral equation Z t Z t xt := E[Xt] = E[X0] + (1 − α) E[Xs]dsE[Ms] = x0 + (1 − α) xse˙sds ; t ≥ 0 . 0 0 Solving this equation, we have xt = E[Xt] = x0 exp((1 − α)et) , t ≥ 0 . Then substituting it into (92), we have Z t Z t Xt = X0 + [(−a)(Xs − xs) − αxs e˙s]ds + Wt + xsdMs ; t ≥ 0 . 0 0 For a given smooth function ϕ : [0, ∞) → R with a compact support so that ϕ(0) = 0 , we apply the change of variable formula for the càdàg semimartingale to obtain Z h Z h 0 0 ϕ(Xt+h) = ϕ(Xt) + ϕ (Xt+u)[(−a)(Xt+u − xt+u) − αxse˙s]du + ϕ (Xt+u)dWt+u 0 0 Z h X 1 00 + [ϕ(Xt+u) − ϕ(Xt+u−)] + ϕ (Xt+u)duhX it+u 2 0≤u 32 from ϕ(Xt+u−) = ϕ(0) = 0 to ϕ(Xt+u) = ϕ(xt+u) . For h > 0 , multiplying by 1/h , taking expectations, letting h ↓ 0 , we obtain Z ∞ Z ∞ Z ∞ 0 1 00 ∂t ϕ(x)p(t, x)dx = ϕ (x)[(−a)(x − xt) − αxte˙t]p(t, x)dx + ϕ (x)p(t, x)dx + ϕ(xt)e ˙t 0 0 2 0 ˙ for t ≥ 0 . Note that since the intensity of jumps in X· occurs at the rate eb· , the last term converges in probability, as h ↓ 0 , 1 X 1 X Z ∞ [ϕ(Xt+u) − ϕ(Xt+u−)] = ϕ(xt+u) −−→ ϕ(xt)e ˙t = ϕ(x)e ˙tδxt (dx) , (93) h h h↓0 0≤u 1 2 ∂tp(t, x) + ∂x[(−a(x − xt) − αxte˙t)p(t, x)] − ∂ p(t, x) =e ˙t δx (dx) (94) 2 xx t R ∞ for t ≥ 0 , x ∈ (0, ∞) with xt = 0 xp(t, x)dx , t ≥ 0 . For the boundary condition it is natural to assume that for every t ≥ 0 lim p(t, x) = 0, lim p(t, x) = 0 , lim ∂xp(t, x) = 0 . (95) x↓0 x→+∞ x→+∞ R ∞ Then since P(Xt ∈ [0, ∞)) = 0 p(t, x)dx = 1 , interchanging the order of differentiation and integration, substituting (94) and integrating once, we obtain ∂ Z ∞ Z ∞ 1 0 = p(t, x)dx = ∂tp(t, x)dx = ∂xp(t, 0) +e ˙t , ∂t 0 0 2 where we used (95) to compute the definite integrals. Thus d 1 e˙t = E[Mt] = − ∂xp(t, 0) , (96) dt 2 and we have the nonlinearity term in (94). B Proof of Theorem 5 Proof. (i) Let us assume that a > 0. Without the Dirac term, the ODE (52)–(54) has the following set of solutions : x 2 Z 2 x 7→ e−a(x−x0) −2x0e0(x−x0) A + B ea(y−x0) +2x0e0(y−x0)dy , (97) 0 for some constants A and B. Hence the solution to the ODE (52)–(54) must have this form before and after x0 with a change of regime at the point x0. We solve separately on each interval [0, x0) and (x0, ∞), and look for a continuous density function p. On the interval [0, x0) : Take a solution of the form of (97). The boundary condition p(0) = 0 0 implies A = 0. The boundary condition p (0) = 2e0 implies B = 2e0. Hence the solution is : x 2 Z 2 −a(x−x0) −2x0e0(x−x0) a(y−x0) +2x0e0(y−x0) p(x) = 2e0e e dy (98) 0 x Z 2 2 ay +2x0(e0−a)(y−x)−ax = 2e0 e dy. (99) 0 33 On the interval (x0, ∞) : Take a solution of the form of (97). If p is continuous, then we must have : x 2 Z 2 −a(x−x0) −2x0e0(x−x0) a(y−x0) +2x0e0(y−x0) p(x) = e p(x0) + B e dy , x0 where p(x0) is given by (98) at point x0. Now the dynamics (52)–(54) indicates formally that : 0 0 p (x0+) − p (x0−) − = e0. 2 Hence, according to (98), we have : 0 0 −2x0e0p(x0) + B = p (x0+) = −2e0 + p (x0−) = −2e0 − 2x0e0p(x0) + 2e0. Therefore B = 0 and X· has a gaussian distribution on the left side of x0, i.e. : 2 −a(x−x0) −2x0e0(x−x0) p(x) = p(x0)e . Global solution : Combining the results on the two intervals, we obtain (55). Characterization of e0 : It remains to derive e0 which satisfies the two relations : Z ∞ Z ∞ xp(x)dx = x0, and p(x)dx = 1. (100) 0 0 R ∞ Let us assume that 0 p(x)dx = 1. Then, one can check by direct computation that : Z ∞ Z ∞ x0 2e0 xp(x)dx = − (e0 − a) p(x)dx + 2 x0 = x0. 0 a 0 τ R ∞ √ Hence e0 is determined by the relation 0 p(y)dy = 1. Using the notation τ = 2a, this relation rewrites : x0 x ∞ x0 1 Z Z 2 2 Z Z 2 2 = e−ax −2x0(e0−a)(x−y)+ay dydx + e−ax −2x0(e0−a)(x−y)+ay dydx 2e0 x=0 y=0 x=x0 y=0 x0 x0 x0 ∞ Z Z 2 2 Z Z 2 2 = e−ax −2x0(e0−a)(x−y)+ay dxdy + e−ax −2x0(e0−a)(x−y)+ay dxdy y=0 x=y y=0 x=x0 x0 ∞ Z 2 Z 2 = eay +2x0(e0−a)y e−ax −2x0(e0−a)xdxdy y=0 x=y x 2 ∞ 2 Z 0 1 τy+ 2x0 (e −a) Z − 1 τx+ 2x0 (e −a) = e 2 τ 0 e 2 τ 0 dxdy y=0 x=y 2x0 2 Z τx0+ τ (e0−a) 2 Z ∞ 1 2x0 1 y − τx+ (e0−a) = e 2 e 2 τ dxdy τ 2x0 1 2x0 y= τ (e0−a) x= τ (y− τ (e0−a)) 2x0e0 ∞ 1 Z τ y2 Z x2 2 − 2 = 2 e e dxdy. τ 2x0e0 y= τ −τx0 x=y Hence e0 is given by (56). Uniqueness of e0 : To conclude the proof, let us show that there is a unique e0 satisfying (56). Let us define the continuous functions : 2x0u ∞ 1 Z τ y2 Z x2 1 2 − 2 F (u) = 2 e e dxdy , G(u) = , 0 < u < ∞ . τ 2x0u 2u y= τ −τx0 x=y 34 and show that there exists e0 ∈ (0, +∞) such that F (e0) = G(e0) . One can check that for every x > 0, Z ∞ x −x2/2 −y2/2 1 −x2/2 2 e ≤ e dy ≤ e . 1 + x x x Using this inequality, we obtain the following lower and upper bounds of F (u) , for every u > τ 2/2 : 2x u 2 2 2 Z 0 1 τ + 4x0u 1 τ x 2 log 2 2 2 = 2 2 dx (101) 2τ τ + (2x0u − τ x0) τ 2x0u 1 + x τ −τx0 2x0u 1 Z τ 1 1 2u ≤ F (u) ≤ 2 dx = 2 log 2 . τ 2x0u x τ 2u − τ τ −τx0 Thus limu→∞ F (u) = 0 , and hence limu→∞(F (u) − G(u)) = 0 . Moreover, given τ > 0 and x0 > 0 , let us choose t1 > 1 / 2 such that : e1/t1 − 1 2ax2 = τ 2x2 > > 0 . (102) 0 0 2 2 1/t1 4t1 − (2t1 − 1) e 2 2 Then F (u1) − G(u1) > 0 at u1 = t1τ > τ / 2 , because (101) and (102) imply : 2 2 2 2 1 1 + 4t1τ x0 1 2 F (t1τ ) ≥ 2 log 2 2 2 > 2 = G(t1τ ) . 2τ 1 + (2t1 − 1) τ x0 2t1τ Combining this observation with the continuity of F (·) − G(·) and the fact that limu→0+(F (u) − G(u)) = −∞ , we claim the existence of e0 ∈ (0, u1) such that F (e0) = G(e0) = 2 / e0 . Indeed, by differentiation of F (·) and by change of variables we obtain : c2 Z ∞ 2 Z ∞ 0 2x0 h 1 −x2/2 (c1−c2) −x2/2 i F (u) = e 2 e dx − e 2 e dx 3 {c =2x u/τ , c =τx } τ c1 c1−c2 1 0 2 0 Z ∞ 2x0 h v v 2 i = − e−v /2dv 3 2 2 1/2 2 2 1/2 τ 0 (c1 + v ) [(c1 − c2) + v ] {c1=2x0u/τ , c2=τx0} and hence, F 0(u) > 0 for 0 < u < τ 2 / 4 and F 0(u) ≤ 0 for u ≥ τ 2 / 4 . Thus the function F (u) is unimodal and takes the unique maximum at u = τ 2 / 4 . Since u 7→ G(u) is monotonically decreasing to zero, F (·) − G(·) is also unimodal with limu→∞(F (u) − G(u)) = 0 . Therefore, the solution e0 to F (u) = G(u) is at most one in (0, 2at1) . Upper bound for e0 in (57). Let us evaluate an upper bound 2at1 for e0 . The condition (102) is equivalent to 2 2 1/t1 2 1/t1 2 4t1(2ax0 − e ) + 4(2ax0)t1 + 1 − e (1 + 2ax0) > 0 . 2 1/t1 2 2 2 This holds if 2ax0 ≥ e and if 4(2ax0)t1 + 1 − e (1 + 2ax0) ≥ 0 , because t1 > 1 / 2 . Thus it 2 2 2 2 suffices to have t1 ≥ 1/(log(2ax0) and t1 ≥ (e (1 + 2ax0) − 1)/(4ax0) . This implies (57). (ii) Let us assume that a = 0. The proof of (58) is very similar to the proof of (55) so we do not reproduce it. As for the characterization of e0, note that the second equality in (100) is always satisfied in this case. The first equality leads to (58). 35 C Proof of (33) With the renewal theory, we shall show the last inequality in (33), namely, ∞ X + t P( sup (Ws + s) ≥ kx0) ≤ ; t ≥ 0 . (103) 0≤s≤t x0 k=1 under the assumption x0 ≥ 1 . Let us denote by ξ the first passage time of Brownian motion with constant drift for the level x0(> 0) , i.e, ξ := inf{s > 0 : Ws + s ≥ x0} , and consider the sequence ξ, ξ1, ξ2,... of independent copies of ξ , the cumulative sum Sn := ξ1 + ··· + ξn and the renewal P∞ process N(t) := k=1 1{Sk≤t} , t ≥ 0 . The density function and the Laplace transform of ξ is well known, e.g., section 3.5.C of Karatzas & Shreve [18]. Then the right hand of (103) becomes ∞ ∞ X + X P( sup (Ws + s) ≥ kx0) = E[1{Sk≤t}] = E[N(t)] . 0≤s≤t k=1 k=1 which satisfies the renewal equation Z t m(t) := E[N(t)] = P(ξ ≤ t) + E[N(s)]P(ξ ∈ ds); t ≥ 0 . (104) 0 Hence applying the Laplace transforms Z ∞ √ −θt −θξ mb (θ) := e m(t)dt , fb(θ) := E[e ] = exp x0(1 − 2θ + 1 ) ; θ > 0 0 to the renewal equation (104), we solve it in terms of Laplace transforms : fb(θ) mb (θ) = ; θ > 0 . θ(1 − fb(θ)) Note that the Laplace transform of `(t) := t / x0 , t ≥ 0 is Z ∞ −θt 1 `b(θ) = e `(t)dt = 2 ; θ > 0 . 0 x0 θ If x0 ≥ 1 , by direct calculations we verify that `b(·) − mb (·) = `\− m(·) is completely monotone: k k d (−1) [`\− m](θ) ≥ 0 ; k ∈ N0 , θ > 0 . (105) dθk Note that if x0 < 1 , then (105) does not hold for some θ and some k ∈ N0 . Thus by Post’s inversion formula (Theorem XIII.10.3 of Feller [11] and also see Theorem 1.4 of Schilling, Song & Vondracek [27]) of Laplace transforms with (105) we conclude that if x0 ≥ 1 , then ∞ − k k+1 k X + ( 1) k d mb P( sup (Ws + s) ≥ kx0) = m(t) = lim k (θ) 0≤s≤t k→∞ k! t dθ θ = k / t k=1 − k k+1 k ( 1) k d `b t ≤ lim k (θ) = `(t) = ; t ≥ 0 . k→∞ k! t dθ θ = k / t x0 This is (103), which completes the proof of (33). 36